## Abstract

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ϵψ(G), if S is a maximum stable set of the subgraph induced by S ∪N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232-248], proved that any S ϵ ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91-101] we have shown that the family ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163-174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414-2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89-94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family ψ(G) is a greedoid if and only if G has a unique perfect matching.

Original language | English |
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Pages (from-to) | 245-252 |

Number of pages | 8 |

Journal | Discrete Mathematics, Algorithms and Applications |

Volume | 3 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 2011 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2011 World Scientific Publishing Company.

## Keywords

- König-Egerváry graph
- Very well-covered graph
- greedoid
- local maximum stable set
- triangle-free graph